Solve for $x$ : $ 3|x - 8| - 3 = -3|x - 8| + 9 $
Explanation: Add $ {3|x - 8|} $ to both sides: $ \begin{eqnarray} 3|x - 8| - 3 &=& -3|x - 8| + 9 \\ \\ { + 3|x - 8|} && { + 3|x - 8|} \\ \\ 6|x - 8| - 3 &=& 9 \end{eqnarray} $ Add ${3}$ to both sides: $ \begin{eqnarray} 6|x - 8| - 3 &=& 9 \\ \\ { + 3} &=& { + 3} \\ \\ 6|x - 8| &=& 12 \end{eqnarray} $ Divide both sides by ${6}$ $ \dfrac{6|x - 8|} {{6}} = \dfrac{12} {{6}} $ Simplify: $ |x - 8| = 2$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 8 = -2 $ or $ x - 8 = 2 $ Solve for the solution where $x - 8$ is negative: $ x - 8 = -2 $ Add ${8}$ to both sides: $ \begin{eqnarray} x - 8 &=& -2 \\ \\ {+ 8} && {+ 8} \\ \\ x &=& -2 + 8 \end{eqnarray} $ $ x = 6 $ Then calculate the solution where $x - 8$ is positive: $ x - 8 = 2 $ Add ${8}$ to both sides: $ \begin{eqnarray} x - 8 &=& 2 \\ \\ {+ 8} && {+ 8} \\ \\ x &=& 2 + 8 \end{eqnarray} $ $ x = 10 $ Thus, the correct answer is $x = 6 $ or $x = 10 $.